Expectation

bobbym · 2012-11-05 02:44:25

Hi;

That is not the shortest way to get the expected value of an absorbing chain. Anyway for your matrix the correct answer is 2.

There is a much simpler general way to get it from all absorbing chains.

anonimnystefy · 2012-11-05 03:11:45

What is that way?

bobbym · 2012-11-05 03:23:48

Without the labels:

This is in the correct form which will be explained as we do larger problems. Anyway, you now throw away all rows that have a 1 and any rows below that. Then you throw away any columns with a 1 and all columns to the right of that one. You get:

As problems get bigger this method will look better and better.

anonimnystefy · 2012-11-05 03:34:01

Well, let's try it on another problem. Got any?

bobbym · 2012-11-05 03:39:56

Hi;

Same type but just a bit more difficult:

How can you compute the expectation of the number of throws for a fair coin until you get a certain pattern, say two heads in a row?

anonimnystefy · 2012-11-05 03:45:09

Tt Th Ht Hh
Tt (1/2 , 1/2 , 0 , 0)
Th ( 0 , 0 , 1/2,1/2)
Ht (1/2, 1/2 , 0 , 0)
Hh (0 , 0 , 0 , 1)

bobbym · 2012-11-05 04:04:04

Hi;

That looks about right, very good. Now you must solve.

anonimnystefy · 2012-11-05 04:22:12

Is the answer 16?

bobbym · 2012-11-05 04:24:16

Hi;

anonimnystefy · 2012-11-05 04:36:26

Maybe 4?

bobbym · 2012-11-05 09:15:43

Hi;

After the rows and columns are eliminated.

The answer is 6.

anonimnystefy · 2012-11-05 10:06:33

I got all that but wasn't sure what to do with the resulting column vector...

bobbym · 2012-11-05 10:22:28

Hmmm, I have trouble with reading the answer too. Know what helps?

anonimnystefy · 2012-11-05 12:03:21

What?

bobbym · 2012-11-05 12:06:28

Crom is strong! If I die, I have to go before him, and he will ask me, 'What is the riddle of steel?' If I don't know it, he will cast me out of Valhalla and laugh at me." -Conan the Barbarian

Experimental math! Back engineering! I almost always have the answer first, or the bounds the answer lies in. That is the secret of the riddle of steel. You are trying to solve them forward to back.

There are numerous other ways to demonstrate the answer of 6 ( notice the use of the word demonstrate ).

I will be busy today and away from the forum. We will continue with more problems when I get back if you want.

anonimnystefy · 2012-11-06 03:45:50

Hi bobbym

How'd you get that equation?

What have you been busy with? The elections?

bobbym · 2012-11-06 03:49:19

Hi anonimnystefy;

In will explain fully how that can be done when I get back from the elections. See you a little later.

anonimnystefy · 2012-11-06 04:00:27

You are actually gonna vote?

anonimnystefy · 2012-11-06 04:22:21

Btw, I found a better Markov chain for the problem:

(1/2 , 1/2 , 0)
(1/2 , 0 , 1/2)
( 0 , 0 , 1 )

With states: empty string, 'H' and 'HH'. These string keep record of what we had till now. If we didn't throw the coin or the last throw was a tail then we have the state string to be an empty string. If the last throw was a head and the state string before the throw was empty, then the new string is 'H'. And finally, if last state string is 'H' and we get another head, then the new state string is 'HH' and this is the absorbing state. When we apply the method you have there, we will get the column vector (6,4) which means that we can get to HH in 6 steps from an empty string and 4 steps from one H. So the answer is 6.

bobbym · 2012-11-06 09:17:17

Hi;

Yes, I vote. Does that surprise you?

Sometimes you find a shorter chain. I know about that one but when someone is trying to learn I do not take any shortcuts. Also, I am not Paul Keres.

Where were we? You wanted to know the derivation of the formula?

anonimnystefy · 2012-11-06 09:52:12

No I didn't. Btw, the formula seems kinda intuitive when compared to 1/(1-x) for summing the geometric series...

A new problem would be nice...

Yes, I'm surprised that you vote. Do you just cancel the ballot?

bobbym · 2012-11-06 10:01:53

I am talking about the formula in post #40.

No, I vote for a candidate. Why are you surprised?

anonimnystefy · 2012-11-06 10:08:44

Oh, that one! Yes, I am curious as to how you got it.

I am surprised because I have no idea who you would vote for...

bobbym · 2012-11-06 10:12:35

Hmmm, most of the time we for against something, not for something. Of course it does not matter but, have you read "The Sea Wolf," by Jack London?

This is what I have in my notes:

Let E denote the expected number of tosses before you get two heads in a row. If your first toss is a tails, then the expected number of total tosses before two heads is E + 1. If your first toss is a head and your second toss is a tail, then the expected number is E + 2. If you toss two heads, then the number of tosses is 2. These events occur with probability 1/2, 1/4, and 1/4, respectively.

anonimnystefy · 2012-11-06 10:16:19

Ahhhh... I get it. Thank you for the formula.

No, I haven't read it.

Math Is Fun Forum

#26 2012-11-05 02:44:25

Re: Expectation

#27 2012-11-05 03:11:45

Re: Expectation

#28 2012-11-05 03:23:48

Re: Expectation

#29 2012-11-05 03:34:01

Re: Expectation

#30 2012-11-05 03:39:56

Re: Expectation

#31 2012-11-05 03:45:09

Re: Expectation

#32 2012-11-05 04:04:04

Re: Expectation

#33 2012-11-05 04:22:12

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#34 2012-11-05 04:24:16

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#35 2012-11-05 04:36:26

Re: Expectation

#36 2012-11-05 09:15:43

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#37 2012-11-05 10:06:33

Re: Expectation

#38 2012-11-05 10:22:28

Re: Expectation

#39 2012-11-05 12:03:21

Re: Expectation

#40 2012-11-05 12:06:28

Re: Expectation

#41 2012-11-06 03:45:50

Re: Expectation

#42 2012-11-06 03:49:19

Re: Expectation

#43 2012-11-06 04:00:27

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#44 2012-11-06 04:22:21

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#45 2012-11-06 09:17:17

Re: Expectation

#46 2012-11-06 09:52:12

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#47 2012-11-06 10:01:53

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#48 2012-11-06 10:08:44

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#49 2012-11-06 10:12:35

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#50 2012-11-06 10:16:19

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